KC Sinha Mathematics Solution Class 11 Chapter 3 Functions Exercise 3.1

Exercise 3.1

page no 3.26

Type 1

Question 1

Examine which of the following relations are functions? If they are functions, find their domain and range :

(i)

$f=\{(1,4),(2,3),(3,4),(4,3)\}$

(ii)

$g=\{(2,5),(-1,0),(1,6)\}$

(iii)

$h=\{(1,2),(2,2),(3,2)\}$

(iv)

$\phi=\{(1,2),(1,3),(2,5)\}$

(v)

$\psi=\{(2,1),(3,1),(5,2)\}$

(vi)

$u=\{(2,1),(5,1),(8,1),(11,1),(14,1),(17,1)\}$

(vii)

$y=\{(0,0),(1,1),(1,-1),(4,2),(4,-2),(9,3),(9,-3),(16,4),(16,-4)\}$

Question 2

Let

$A=\{1,2,3\}$ and

$B=\{4,5\}$ .

Let

$f=\{(2,4),(3,5)\}$ . Is

$f$ a function from

$A$ into

$B$ (from

$A$ to

$B$ ).

Question 3

Let

$A=\{1,2,3\}, B=\{3,6,9,10\}$ . Which of the following relations are functions from

$A$ to

$B$ ? Also find their range if they are functions.

$f=\{(1,9),(2,3),(3,10)\}$

$g=\{(1,6),(2,10),(3,9),(1,3)\}$

$h=\{(2,6),(3,9)\}$

$u=\{(x, y): y=3 x, x \in A\}$

Page no 3.27

Question 4

Let

$A=\{1,2,3\}, B=\{4,5\}$

Let

$f=\{(1,4),(1,5),(2,4),(3,5)\}$ . Is

$f$ a function from

$A$ to

$B$ ?

Question 5

Let

$A=\{a, b, c, d\}$ . Examine which of the following relations is a function on

$A$ ?

(i)

$f=\{(a, a),(b, c),(c, d),(d, c)\}$

(ii)

$g=\{(a, c),(b, d),(b, c)\}$

(iii)

$h=\{(b, c),(d, a),(a, a)\}$

Question 6

Let

$A=\{1,2,3,4\}, B=\{1,5,9,11,15,16\}$

and

$f=\{(1,5),(2,9),(3,1),(4,5),(2,11)\}$

$f$ a function from

$A$ to

$B$ .

Question 7

(i) Let

$f$ be the subset of

$Q \times Z$ defined by :

$f=\left\{\left(\frac{m}{n}, m\right): m \in Z, n \in Z, n \neq 0\right\}$ . Is

$f$ a function from

$Q$ to

$Z$ ? Justify your answer.

(ii) The relation

$f$ is defined by

$f(x)= \begin{cases}x^{2}, & 0 \leq x \leq 3 \\ 3 x, & 3 \leq x \leq 10\end{cases}$

The relation

$g$ is defined by

$g(x)= \begin{cases}x^{2}, & 0 \leq x \leq 2 \\ 3 x, & 2 \leq x \leq 10\end{cases}$

Show that

$f$ is a function and

$g$ is not a function.

[Hint :

$g(2)=2^{2}=4$ and

$g(2)=3 \times 2=6 \quad \therefore 2$ has two images 4 and 6 ]

Question 8

Let

$N$ be the set of natural numbers and the relation

$R$ be defined on

$N$ such that

$R=\{(x, y): y=2 x, x, y \in N\}$ . Find the domain, codomain and range of

$R$ . Is this relation a function.

Type 2

Question 9

(i) A function

$f$ is defined by

$f(x)=2 x-5$ , find

$f(0), f(7)$ and

$f(-3)$

(ii) If

$f(x)=x^{2}+4, x \in R$ then, find

$f(2), f(-5)$ and

$f(t+2), t \in R$

(iii) If

$f(x)=x^{2}+1, x \in R$ , find

$f(1) \times f(4)$

Question 10

$f: R \rightarrow R$ be defined by

$f(x)=x^{2}+2 x+1$ . Then find :

(i)

$f(-1) \times f(1)$ Is

$f(-1)+f(1)=$

$f(0) ?$ (ii)

$f(2) \times f(3)$ . Is

$f(2) \times f(3)=f(6) ?$

Question 11

(i) Let

$f=\{(1,1),(2,3),(0,-1),(-1,-3)\}$ be a function from

$Z$ to

$Z$ defined by

$f(x)=a x+b$ for some integers

$a, b$ , determine

$a$ and

$b$ .

(ii) Let

$f=\{(1,1),(2,3),(0,-1),(-1,-3)\}$ be a linear function from

$Z$ to

$Z$ , find

$f(x)$ .

Question 12

Which of the following relations are functions ? Give reasons. If it is a function, determine its domain and range.

(i)

$f=\{(2,1),(3,1),(4,2)\}$

(ii)

$g=\{(1,3),(1,5),(2,5)\}$

(iii)

$h=\{(2,2),(2,4),(3,3),(4,4)\}$

(iv)

$\varphi=\{(2,1),(5,1),(8,1),(11,1),(14,1),(17,1)\}$

Type 3

Question 13

Function

$f$ is given by

$f=\{(4,2),(9,1),(6,1),(10,3)]$ . Find the domain and range of

$f$ .

Question 14

Let

$A={9,10,11,12,13}$ and let

$f: A \rightarrow N$ be a function defined by

$f(n)=$ the highest prime factor of

$n$ . Find the range of

$f$ .

Question 15

$A=\{-3,-2,-1,0,1,2,3\}$ and

$f^{\prime}(x)=x^{2}-1$ defines

$f: A \rightarrow R$ . Then find range

$f$

Question 16

Find the domain and range of the following functions.

(i)

$f(x)=x$

(ii)

$f(x)=(x-1)$

(iii)

$f(x)=x^{2}-1$

(iv)

$f(x)=x^{2}+2$

(v)

$f^{\prime}(x)=\sqrt{x-1}$

Question 17

Find the domain of the function

$f(x)=\frac{x^{2}+3 x+5}{x^{2}-5 x+4}$

Question 18

Find the range of the function

$f(x)=2-3 x, x>0$

Question 19

Find the domain and range of the following functions:

(i)

$\left\{\left(x, \sqrt{9-x^{2}}\right): x \in R\right\}$

(ii)

$\{(x,-|x|): x \in R\}$ .

Question 20

Find the domain of definition and range of the function defined by the rules:

(i)

$f(x)=x^{2}$

(ii)

$g(x)=|x|$

(iii)

$h(x)=\frac{1}{3-x^{2}}$

(iv)

$u(x)=\sqrt{4-x^{2}}$

Question 21

Consider the following rules :

(i)

$f: R \rightarrow R: f(x)=\log _{c} x$

(ii)

$g: R \rightarrow R: g(x)=\sqrt{x}$

(iii)

$h: A \rightarrow R: h(x)=\frac{1}{x^{2}-4}$ , where

$A=R-\{-2,2\}$

Which of them are functions? Also find their range, if they are functions.

Question 22

Let

$f: R-\{2\} \rightarrow R$ be defined by

$f(x)=\frac{x^{2}-4}{x-2}$ and

$g: R \rightarrow R$ be defined by

$g(x)=x+2$ Find whether

$f=g$ or not.

Type 4

Question 23

Let

$f: R \rightarrow R$ and

$g: R \rightarrow R$ be defined by

$f(x)=x+1, g(x)=2 x-3$ .

Find

$f+g, f-g$ and

$\frac{f}{g}$

Question 24

Let

$f=\{(1,1),(2,3),(0,-1),(-1,-3)\}$ be a linear function from

$Z$ into

$Z$ and

$g(x)=x$ . Find

$f+g$ .

Question 25

Find

$f+g, f-g, f \cdot g, \frac{f}{g}$ and

$\alpha f(\alpha \in R)$ if

(i)

$f(x)=\frac{1}{x+4}, x \neq-4$ and

$g(x)=(x+4)^{3}$

(ii)

$f(x)=\cos x, g(x)=e^{x}$

(iii)

$f(x)=\sqrt{x-1}, g(x)=\sqrt{x+1}$

Question 26

$f(x)=x, g(x)=|x|$ , find

$(f+g)(-2),(f-g)(2),(f \cdot g)(2)$ ,

$\left(\frac{f}{g}\right)(-2), 5 f(2)$

Type 5

Question 27

Draw the graph of the following functions

(i)

$f: R \rightarrow R$ such that

$f(x)=x-1$

(ii)

$f: R \rightarrow R$ such that

$f(x)=|x-1|$

(iii)

$f: R \rightarrow R$ such that

$f(x)=|x-2|$

(iv)

$f: R \rightarrow R$ such that

$f(x)=4-2 x$

Question 28

Let

$N$ be the set of all natural numbers.

Define a real valued.

function

$f: N \rightarrow N$ by

$f(x)=2 x+1$ .

Using this definition complete the table given below :

$\begin{array}{|cccccccc|}\hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline f(x) & f(1)= & f(2)= & f(3)= & f(4)= & f(5)= & f(6)= & f(7)= \\\hline\end{array}$

Question 29

Define the function

$f: R \rightarrow R$ by

$y=f(x)=x^{2}$ . Complete the table given below :

(Image to be added)

Find the domain and range of function

$f$ and draw its graph.

Question 30

Define the real valued function on

$f: R-\{0\} \rightarrow R$ as

$f(x)=\frac{1}{x}$ Complete the figure given below:

(Image to be added)

Find the domain and range of

$f$ .

KC Sinha Mathematics Solution Class 11 Chapter 3 Functions Exercise 3.1

Exercise 3.1

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